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Dynamical Systems



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What is P5?

P5 is a computer program which studies a specific Piecewise Polynomial Planar Vector Field (or Differential System) of any degree.  Part of its calculations is done through an interface with the computer program Maple (giving the option of treating a vector field using symbolic manipulations), and part of its calculations is done in P5 itself.

A piecewise vector field is a finite number of vector fields defined on a domains that are bounded by a finite number of algebraic curves (bifurcation lines).  Such vector fields are often found in control theory.  The piecewise polynomial vector fields can be discontinuous along the bifurcation lines.

P5 is an extension of P4, and is introduced to be able to treat such piecewise vector fields.  Much of the functionality of P5 is inherited by P4:

 Functionality of P4/P5

  • The program draws the phase portrait on a compact part of the plane, on the Poincaré sphere or on one of the charts at infinity.  The ability to draw on the Poincaré sphere and hence to present a view of the phase portrait on the global compactified phase space is one of the novelties of this program.
  • The program Determines all singular points (finite and infinite ones) of the vector field.  P5 is able to present a detailed study of each singularity.  It distinguishes between the different type of singularities (saddle, node, focus, center, semi hyperbolic saddle-nodes, degenerate).  Singularities with purely imaginary eigenvalues are studied by means of Lyapunov coefficients to distinguish (in some cases) between center behaviour and weak focus behaviour.
  • Degenerate singularities are studied by means of quasi-homogeneous blow-up.  It is shown that a finite number of such blow-ups unfolds the singularity in a number of hyperbolic or semi-hyperbolic singularities, each of which are studied in detail by the program.
  • For each singularity, invariant separatrices are calculated (formally, up to a degree that can be specified).  The formal expansion is tested for its numeric accuracy.  
  • A detailed report can be prepared containing a list of singularities, their separatrices, their sectors (parabolic, hyperbolic or elliptic).  The program can plot all or individual separatrices.  To speed up integration of the separatrices of degenerate singularities (where the flow is generally very slow), it uses the rescaled (blown up) versions of the vector field in a neighbourhood of the singular point.
  • All the above calculations can be done either in numeric mode (with given precision), or in symbolic mode (using Maple as a symbolic math manipulator).
  • The program does not deal with bifurcations.  The program does offer the possibility of introducing a vector field with user-defined parameters.  Each time you want to study the vector field (examine singularities/draw phase portraits), you just have to enter the value of the parameters.
  • The program is able to find periodic orbits up to a certain degree of precision (in particular limit cycles, i.e. isolated periodic orbits), cutting a transverse section that is specified by the user.
  • Export images to XFIG, Encapsulated Postscript, or JPEG.

Functionality of P5

  • P5 calculates the domains bounded by algebraic curves (up to some precision).  These bifurcation lines are drawn on the phase portrait.
  • P5 studies singularities of the several vector fields that are used to define a piecewise polynomial vector field.  Singularities that lie on a bifurcation line deserve special attention: singularities of mixed type are possible, and only those separatrices that start in the relevant domain are to be drawn on the phase portrait.
  • A integration parameter allows to control the integration of orbits near bifurcation lines.  Phenomena such as sliding vector fields and sewing can be observed when using the program.